Normality concerning exceptional functions

Abstract

Let $\varphi(z)(\not\equiv0)$ be a function holomorphic in a domain $D$, $k\in\mathbb{N}$, and let $\mathcal{F}$ be a family of meromorphic functions defined in $D$, all of whose zeros have multiplicity at least $k+2$ such that, for every $f\in\mathcal{F}$, $f^{(k)}(z)\neq\varphi(z)$. The non-normal sequences in $\mathcal{F}$ are characterized.